“Is the baby's placenta impure or najis? If so,
does it belong to the najis mutawassitah?”

Far from it, the placenta is pure and it is not an impurity. (However, it
goes without saying that blood found together with the placenta during
childbirth is impure.) The best reference for this can be found in the
commentary to our very first textbook on Fard 'Ayn, the Safinat al-Naja,
where the "Second Nawawi" (or more fondly, honoured in Arabic as "al-Nawawi
al-Saghir"), Imam Nawawi al-Bantani al-Jawi (may Allah be pleased with
him!) says:

"The placenta [mashima] that comes out with the child is pure." [al-Nawawi
al-Jawi, Kashifat al-Saja, 43].

Know that our fuqaha' divide the placenta into 2 types: (1) the normal
placenta surrounding the baby, that is to say, the 'wrapping wall'
[al-mashima allati fi-ha al-walad]; and (2) the umbilical cord [al-khalas].
Both types are pure. However, because the latter type is considered to be a
part of the baby (made clear by Imam al-Qalyubi and verified by the Shaykh
al-Azhar, Imam al-Birmawi (as recorded by al-Shirwani and al-Bujayrmi (may
Allah be pleased with them all!) [al-Shirwani, Hawashi, 4;125]); cf.
al-Qalyubi, Hashiyatan, 1:395; while al-Birmawi's original 'ibara is in his
as of yet unpublished Hashiya of Fath al-Wahhab, not in his Hashiya of Fath
al-Qarib), it is therefore Mandub, Sunna or recommended to bury the khalas
in the ground if the baby is alive. That one should bury it in the ground
is made unequivocally by the Mufti of the Shafi'is in Makka, Sayyid Zayni
Ahmad Dahlan (may Allah be pleased with him!), in one of his Fatwas
[Dahlan, Muhimmat al-Nafa'is, 19]. (On the other hand, if the baby is dead,
then it becomes Wajib or obligatory to bury it with the rest of the body.)
This is because Allah has honoured Bani Adam, and since the umbilical cord
is also a part of the human body, then it should also be honoured. As is
customarily the practice in many parts of the Muslim world, there is
nothing wrong if it is first cleansed (from impurities such as washing away
the blood), and then wrapped, before it is finally buried. In fact, it is
better to do this than to, literally, 'throwing it to the wolves'. (As for
the text of the Kashifat al-Saja which goes on to say (here, as in the
above passage, al-Nawawi al-Jawi was relying originally on al-Sharqawi's
Hashiya): "[Imam 'Ali] al-Shabramallisi (may Allah be pleased with him!)
said: It is clear [to me; i.e., al-Shabramallisi, the teacher of
al-Birmawi] that it is not required to do anything with the placenta," is
in fact referring to the opinion (with respect to a khilaf in this issue in
our school and the discussions concerning it) that it is not required to
prepare the Mashima (by making Ghusl and shrouding it, etc.) in order to
perform a Janaza prayer for it, if found on its own, like the rest of the
body parts. [Original 'ibara: al-Shabramallisi, Hashiya, 2:493-494; cf.
al-Sharqawi, Hashiyat al-Tahrir, 1:121].

+Fa'ida+ (an extra and interesting note for students of fiqh here):
although the two types of mashima are pure, they are nevertheless, legally
distinct and are both prohibited for human consumption. The fact that it is
Haram to consume them is not because they are impure, but its prohibition
is due to some other legal basis ['illa] (since according to Shafi'i
jurists, the more universal and general definition of Najasa (note that
there is another, but more particular definition which is omitted here) is,
"every thing that is prohibited to consume, (1) unrestrictedly [whether a
little or a lot], (2) when having an option, (3) as well as being easily
distinguishable, (4) whereas it is not [prohibited to consume] due to it
being sacred, (5) nor due to it being disgusting, and (6) nor due to it
being harmful to the body and mind." [al-Bajuri, Hashiya, 1:99-100; note,
only the fourth and fifth qualifiers of this legal definition are relevant
to us here]). Thus on the one hand, it is Haram to consume the umbilical
cord, (and the prohibition of its consumption is not due to it being an
impurity but) because it is considered sacred or 'Hurma', like a human
corpse (and whether a Muslim's or a non-Muslim's it is still pure). On the
other hand, it is Haram to consume the baby's placenta, (and again, with
exception from the legal reason that it is an impurity but) because it is
considered repulsive or 'Istiqdhar', just like the mucus, saliva and sperm
(which are all, however, legally pure). These exceptive categories of Hurma
and Istiqdhar (together with the two other exceptions of pure things,
namely: what is harmful to the body [Darar fi badan], such as stones and
earth, and what is harmful to the mind [Darar fi 'aql], such as cannabis
and hashish) are what excludes them from being defined a Najasa, and what
qualifies that general definition.

Someone seeking the pleasure of Allah have composed the following [min bahr
al-kamil]:

[Behold, all of the placenta is pure,
whereas only its blood is impure.
As for the umbilical cord, it is Haram [to eat] owing to it being sanctified,
while the other [is Haram] in view of it being foul.
The legal reason of Haram is not because of impurity
but instead because it is sanctified.
Thus you should bury the placenta,
like the rest of mankind, who are honoured.]